Watch the end of the Buffy episode with the invisible girl... or 99% of the X-files episodes... that is what would happen. The government would make up a nice story if needed, but they would lock the person up, try to find out what gives him/her those powers, if they can reproduce it in others, how they can use it for their own benefit (with brainwashing if needed), and how to keep it out of the hands of the enemy. If you refuse to cooperate with them they prolly consider you too big a risk to let you go, and will kill you before letting you run around loose. They'd rather see you dead than risk an enemy getting their hands on you and finding out how to use your powers for his/her own purpose.
Edit: Imagine the inventors of the atom bomb publishing some, but not all of the information needed to make a bomb, and explaining the destructive powers of it, but not going to a government with it. Within minutes after the release of such a document there would have been special marines on the way from the US, Canada, the UK, Russia, Germany, Italy and Japan. Some knowledge and power should remain hidden, for whatever side gets its hand on it will use it for their own benefit, paving the way for even more terrible powers.
